Abstract

In this article, we prove that any complex smooth rational surface X which has no automorphism of positive entropy has a finite number of real forms (this is especially the case if X cannot be obtained by blowing up \(\mathbb P^2_{\mathbb C}\) at \(r\ge 10\) points). In particular, we prove that the group \(\mathrm {Aut\,}^{\#}X\) of complex automorphisms of X which act trivially on the Picard group of X is a linear algebraic group defined over \(\mathbb R\).

Mathematics Subject Classification

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Acknowledgments

The author is grateful to Frédéric Mangolte for asking him this question, and also for his advice. We want to thank Jérémy Blanc, Serge Cantat, Stéphane Druel, Viatcheslav Kharlamov and Stéphane Lamy for useful discussions.